Curvature-induced stiffening of a fish fin Khoi Nguyen1, Ning Yu2,**, Mahesh M. Bandi1, Madhusudhan Venkadesan3, and Shreyas Mandre4,* 1Collective Interactions Unit, OIST Graduate University, Onna, Okinawa, 904-0495, Japan 2Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, P. R. China 3Department of Mechanical Engineering and Materials Science, Yale University, New Haven, CT 06520, USA 4School of Engineering, Brown University, Providence, RI 02912, USA *corresponding author: shreyas [email protected] **research performed while visiting Brown University Abstract How fish modulate their fin stiffness during locomotive manoeuvres remains unknown. We show that changing the fin’s curvature modulates its stiffness. Modelling the fin as bendable bony rays held together by a membrane, we deduce that fin curvature is manifested as a misalignment of the principal bending axes between neighbouring rays. An external force causes neighbouring rays to bend and splay apart, and thus stretches the membrane. This coupling between bending the rays and stretching the membrane underlies the increase in stiffness. Using 3D reconstruction of a Mackerel (Scomber japonicus) pectoral fin for illustration, we calculate the range of stiffnesses this fin is expected to span by changing curvature. The 3D reconstruction shows that, even in its geometrically flat state, a functional curvature is embedded within the fin microstructure owing to the morphology of individual rays. Since the ability of a propulsive surface to transmit force to the surrounding fluid is limited by its stiffness, the fin curvature controls the coupling between the fish and its surrounding fluid. Thereby, our results provide mechanical underpinnings and morphological predictions for the hypothesis that the spanned range of fin stiffnesses correlates with the behaviour and the ecological niche of the fish. Keywords: fish fin; biomechanics; curvature 1 Introduction Ray-finned fish, or Actinopterygii, use their fins as a general purpose device to manipulate the surrounding fluid. That the rayed structure of the fin is found in greater than 99% of all living fish species is a testament to its versatility in the aquatic environment [11, 16, 19]. The fin is used in a wide range of manoeuvres [30], from holding stationary in spatio-temporally varying turbulent flows to explosive bursts of motion. Insight into design principles underlying its biomechanics could therefore shed light into its overwhelming dominance in the aquatic realm, and find applications in underwater robotics. The mechanical stiffness of the fin is a key parameter that influences its propulsive performance [2, 6, 23, 26, 29, 34]. What structural elements might be responsible for the fin’s stiffness? The fin stiffness under bending deformations is tacitly assumed to equal the individual ray bending rigidity multiplied by the arXiv:1608.05891v3 [physics.flu-dyn] 1 May 2017 number of rays [21]. Experimentally, an individual ray's bending rigidity is estimated by supporting it in a cantilevered loading and measuring the displacement at a point where a force is applied to deform the ray [1, 9]. Here we show using a mathematical model that fin curvature along a direction transverse to the rays stiffens it to bending beyond this simple picture. The mechanism of this stiffening underlies the common observation that, under its own weight, a flat sheet of paper droops but a slightly curved sheet stiffens and barely deforms. Curvature couples out-of-plane bending of the sheet to its in-plane stretching; the in-plane elastic properties of the sheet thus become relevant in determining its stiffness. However, unlike a sheet of paper, rayed fins are composite structures consisting of hard bony rays interconnected by softer membranes (a) (b) longitudinal membranes transverse Fin base rays membrane ray transverse x longitudinal L Fin base Figure 1: Schematics for a model fish fin. (a) A prototypical fish fin has multiple bony rays which attach to a common fin base and are held together by tissue membranes. (b) We model the fin using elastic beams for rays and distributed springs for membranes. The rays and membranes are of length L and approximate a transverse curvature by tracing a constant circular arc. [15, 42]. We develop a simple mathematical model that encapsulates the essence of the bending-stretching coupling applied to the rayed-fin structure; bending of the rays is mechanically coupled to the stretching of the membranes. Via this mechanism, the fish can modulate the stiffness of the fin by merely changing its transverse curvature. We calculate the range of stiffnesses spanned by a model fin as a function of curvature and find the important underlying parameters to be the anisotropy in the bending modulus of the rays and the elastic properties of the membrane under stretching. Finally, an important result from our model is the definition of the discrete curvature itself. We identify it to be proportional to the misalignment of the principal axis of bending. By applying this definition, we identify a \functional" curvature although the fin may be geometrically flat. We apply our model to a sample fish fin { a Mackerel (Scomber japonicus) pectoral fin { in which we predict the degree of curvature-induced stiffening and illustrate the presence of a functional curvature. We emphasize that the Mackerel fin was not chosen for a biological function it specializes in. It merely serves to illustrate the mechanical principles, which are the focus of this article. 2 Mathematical model A fish fin has many features that complicate its elastic behaviour. To list a few, individual rays have non- uniform length and heterogeneous internal structures, and the tissue membranes have nonlinear elasticity (see figure 1(a)). To understand the underlying mechanics and isolate the essential principle of bending- stretching coupling, we make the following simplifying assumptions. The membranes are assumed to be identical in composition and dimensions, and resist stretching transverse to the rays with a distributed elastic force. At every distance x from the fin base, the distributed force equals the amount of stretching multiplied by a spring constant k. The fin rays are approximated as linearised Euler-Bernoulli beams with bending rigidities Bn and Bt along principal axes that are aligned with the normal and tangent to the fin surface, respectively. The material of the fin rays and membrane is assumed to be linearly elastic, i.e., the local strain in the material is linearly proportional to the local stress. Because of the transverse curvature, the principal bending directions of each ray is misaligned with its neighbours. A schematic of the fin is shown in figure 1(b) where bending-stretching coupling acts parallel to the cross-section shown (dashed box). With these assumptions, we construct a fin model with few parameters, yet maintain a meaningful approximation to the curvature-induced stiffening of rayed fins. 2 (a) (d) Stiffness modulation γ = 10 between two limits 3B Fin base f R = n (1 + γ) ∞ L3 ) L ( f γ = 5 /w f x = R L γ = 1 3B R = n 0 L3 Stiffness modulation (b) (c) ) (e) 0 as interpolation d4v R between limits 10 100 1000 1 Bt − v(x) dx4 = ∞ line of symmetry ε0 + ε(x)/2 γ R ( / ) kε θ 0 w(x) R − 1/3 R L/l = γ ε0 d4w ( B = n dx4 α kinematics force balance 0.1 1 L/l 10 Figure 2: Results from the analysis of stiffening by curvature in a 2-ray fin. (a) Two rays that are symmetric about a membrane. The rays deform under point forces f applied at the ray tips along their soft principal bending directions. (b) A cross-sectional view of the 2-ray fin at a distance x from the fin base. The displacement of a ray due to point force loadings is decomposed into its principal components, w(x) and v(x). The membrane stretches by "(x) due to the displacement. (c) Forces acting on the ray. θ is the angle the principal axes form with the membrane, Bt and Bn are the tangential and normal bending rigidities respectively, and k is the membrane spring constant. (d) Fin stiffness R transitions between two asymptotic limits as a function of the coupling parameter L=l. The larger the value of γ, the greater the maximal stiffness that can be achieved by an infinitely stiff membrane (L=l ). (e) Rescaling stiffness as an interpolation shows that γ also acts to delay the transition to stronger coupling,! 1 which is characterized by a dashed black line. 2.1 Bending-stretching coupling in a 2-ray fin The basic unit of bending-stretching coupling is a pair of rays connected by a single membrane, a 2-ray fin (figure 2a). A fin with multiple (say N) rays is simply a collective set of 2-ray units, and an exact account of the analogy between a N-ray and a 2-ray fin is presented in Appendix A. Figures 2(a{c) depict the kinematics and balance of forces for a 2-ray fin at a distance x from the fin’s base. Each ray has a hard and a soft principal bending direction, depicted as an anisotropic rectangular cross-section. The misalignment θ between the principal directions and the membranes encodes the fin’s transverse curvature, and defined such that the hard and soft directions are tangential and normal to the fin surface, respectively. That is to say, the membrane surface differs from the fin surface, and this becomes apparent in the N-ray fin of figure 3(a) where the fin surface at any particular ray is the average orientation of its two neighbouring membranes. The 2-ray fin deforms due to an externally applied force causing the rays to bend with displacements w(x) and v(x) along the soft and hard bending directions, respectively.